In EP 1 064 848, the tissue surrounding a vent of poultry is cut loose in a method that can include introducing the centering pin with the second end portion into the vent, rotating the cutter for cutting loose the tissue surrounding the vent, and advancing the clamping sleeve for clamping the cut loose tissue between the clamping sleeve and the thickened end portion of the centering pin, and retracting the centering pin and the clamping sleeve while clamping said tissue for severing the vent from the poultry. In this known method and vent cutter, the centering pin and the clamping sleeve are rotationally fixed as opposed to the rotatable cutter. The cutter can be rotated and advanced to the poultry's tissue for performing a cutting operation on the tissue. Apart from the forward cutting edge of the rotary cutter, the rotary cutter is usually also provided with slits in its circular wall that act on the tissue of the poultry during rotation of the cutter. Further, the clamping sleeve and the cutter of the known vent cutter are in a fixed longitudinal position in relation to each other, which means that when moving the cutter, the clamping sleeve moves in unison with the cutter.